Glossary term
Magnitude-Squared Coherence
Normalized spectral metric between 0 and 1 that indicates how consistently one measured signal is linearly related to another at each frequency.
Definition
metricMagnitude-squared coherence is a normalized spectral metric that estimates how consistently two signals are linearly related at each frequency.
In vibration testing, modal analysis and system identification, magnitude-squared coherence is commonly used to assess the quality of a frequency response function between an excitation signal and a response signal. Values near 1 indicate a strong repeatable linear relation at that frequency; low values can indicate noise, leakage, nonlinearity, multiple unmeasured inputs, poor excitation, sensor problems or inconsistent averaging.
Magnitude-squared coherence measures how consistently two signals are linearly related at each frequency. In modal testing and vibration measurement, it is often used to check whether a measured frequency response function is trustworthy at a given frequency line.
For input signal x(t) and output signal y(t), the magnitude-squared coherence is:
where G_{xy} is the cross-spectrum between x and y, and G_{xx} and G_{yy} are the input and output autospectra. The value is dimensionless:
Values near 1 suggest that the output at that frequency is repeatably and linearly related to the measured input. Values near 0 suggest that the measured input does not explain the output well at that frequency.
Engineering Role
Magnitude-squared coherence is a quality metric, not a physical response by itself. It helps engineers judge whether an FRF peak, anti-resonance, phase trend, damping estimate or fitted modal parameter is supported by consistent input-output data.
The metric is especially useful because a frequency response function can look plausible even when part of the data is unreliable. Coherence gives a frequency-by-frequency warning that the measured input, measured output, estimator and test setup may not support the interpretation. It does not identify the cause by itself.
High coherence is useful, but it does not prove that the test is correct. A wrong sensor calibration, incorrect force direction, bad boundary condition, fixture error or repeatable nonlinear response can still produce apparently coherent data.
Spectral Averaging and Estimation
In practice, the spectra used in coherence are estimated from windowed and averaged records. A simplified averaged cross-spectrum can be written as:
where N_{avg} is the number of averages, X_k and Y_k are Fourier spectra from record k, and * denotes complex conjugation. Autospectra are estimated in the same averaging process:
More averages can reduce random scatter in the estimate, but they do not remove systematic errors. Averaging many bad hammer hits, clipped channels or nonstationary records can create a stable-looking but misleading coherence curve. The averaging method, window, overlap, trigger quality and frequency resolution therefore belong in the test record.
Diagnostic Interpretation
Low coherence can be caused by:
- low signal-to-noise ratio;
- poor excitation energy at that frequency;
- leakage from inadequate windowing or non-periodic data;
- nonlinear behaviour, looseness, friction or saturation;
- unmeasured inputs or multiple simultaneous force paths;
- double hits in impact-hammer testing;
- clipping, sensor overload, weak mounting or cable motion;
- insufficient averaging or poor frequency resolution.
Thresholds are context-dependent. A modal survey might flag \gamma^2<0.90 near an important resonance, while an operating measurement or field diagnostic may accept lower values if the uncertainty is documented. A project-specific acceptance statement can be written as:
That rule is useful only when the decision band, signals, estimator, averaging and consequence of rejection are stated. A single fixed threshold should not replace engineering diagnosis.
Worked Example: Compute Coherence from Spectra
During an impact test at 120\ \text{Hz}, the averaged spectra at one frequency line are:
and the cross-spectrum magnitude is:
The magnitude-squared coherence between force and acceleration is:
Engineering comment: a coherence of 0.810 indicates a fairly strong linear input-output relation, but it is not pristine. The engineer should inspect hammer hits, response level, averaging, windowing and whether other inputs are exciting the structure near 120\ \text{Hz}. If the local acceptance rule required \gamma^2\geq0.90 near this mode, this frequency line would need investigation or retesting.
If the main problem is uncorrelated output noise in a simple single-input case, an approximate signal-to-noise interpretation is:
For \gamma^2=0.810:
or:
For comparison, a coherence threshold of 0.90 corresponds to the simple screen:
This SNR estimate is only a diagnostic screen. It assumes a simple linear single-input situation and should not be used as a universal correction.
Validation and Common Mistakes
A defensible coherence review states signals compared, sensor locations, excitation type, estimator, windowing, averaging, overlap, frequency resolution, input energy, calibration, operating condition and the acceptance threshold used for interpretation.
Coherence should be interpreted alongside the FRF, phase, input spectrum, response level, time records, repeatability and physical plausibility. Low coherence at an anti-resonance may be expected because response is small and noise-sensitive. Low coherence at a resonance that drives a release decision is more serious. High coherence at a wrong boundary condition is still a wrong test.
Common mistakes include:
- treating high coherence as proof of a correct model;
- discarding low-coherence data without identifying whether the cause is noise, nonlinearity, leakage or missing input paths;
- averaging too little and then over-interpreting random coherence variation;
- ignoring coherence at anti-resonances, where response can be small and noise-sensitive;
- comparing coherence values from different estimators or averaging settings;
- using one fixed coherence threshold without considering frequency band, test purpose and measurement consequence.
Distinction from Related Terms
Magnitude-squared coherence is not a frequency response function. An FRF gives response per input as a complex quantity. Coherence indicates how consistent that input-output relation appears at each frequency.
Magnitude-squared coherence is not signal-to-noise ratio. SNR compares signal power with noise power. Coherence compares normalized cross-spectral consistency between two measured signals.
Magnitude-squared coherence is not optical coherence. Optical coherence describes phase correlation in electromagnetic waves. Magnitude-squared coherence here is a signal-processing metric for measured data.
Magnitude-squared coherence is not cache coherence. Cache coherence is a computer architecture property. The shared word “coherence” should not be transferred across domains without context.
Magnitude-squared coherence is not validation by itself. It supports data-quality review, but validation still needs physical plausibility, calibration, repeatability, boundary-condition review and uncertainty assessment.